Performance Analysis of Optical Burst Switching
Networks with and without Class Isolation
Neil Barakat and Edward H. Sargent
Edward S. Rogers School of Electrical and Computer Engineering
University of Toronto
10 King’s College Road
Toronto, Ontario, Canada M5S 3L1
Email: neil.barakat@utoronto.ca, ted.sargent@utoronto.ca
Abstract— This paper presents an analytical model that evaluates the blocking probability of each service class in optical
burst-switching networks. The model is applicable to systems
with arbitrary burst length distributions and arbitrary-sized QoS
header offsets. Thus, unlike previous models, it is applicable to
the design and study of networks with a wide range of traffic
characteristics, including systems in which higher classes are not
necessarily isolated from lower classes and systems in which the
conservation law does not necessarily hold. We derive explicit
expressions for blocking probability both the cases of constant
burst lengths and exponentially distributed burst lengths and
verify the model’s accuracy through simulation. We show the
model to be accurate for a number of different traffic loads
and class priorities. For an OBS system with two classes and
a 1:10 ratio of high-priority to low-priority traffic, our model
is able to predict accurately the blocking probability for each
class, whereas the predictions from a model that assumes isolation
deviates by as much as an order of magnitude from the simulation
results for the higher priority class.
I. I NTRODUCTION
Emerging WDM technology allows huge amounts of data
to be carried transparently in optical networks. Optical burst
switching (OBS) is a means of accommodating bursty traffic
and allowing sub-wavelength granularity in a transparently
optical manner without the need for buffering or other complex functionality in the optical plane. This is achieved by
decoupling of the control plane and the data plane. In OBS,
IP packets are electronically aggregated at the edge of the
network into bursts that are transmitted transparently through
the network core. The core switches are configured in advance
to accept the burst using control packets, which are transmitted
before the bursts in an electronic control plane.
OBS is characterized by two main qualities. First, in order
to accommodate the bounded, non-zero electronic processing
time of control packets in core switches, OBS introduces a
time offset between each burst’s control packet and payload.
This offset eliminates the need for buffering of bursts in the
core of the network, which is typically required in optical
packet-switched networks. Secondly, because of the relatively
short durations of the burst compared to the network size,
most OBS architectures uses one-way reservation schemes in
order to maximize bandwidth efficiency and reduce latency.
In such schemes, the burst is sent into the network without
waiting for a reservation acknowledgement. In one such signal-
ing protocol, called just-enough-time (JET) [1], each control
packet carries information about the offset size and the length
of the burst and attempts to reserve a corresponding bandwidth
window. If such a window is not available due to previous
reservations, the burst is dropped upon arriving at the switch.
An additional offset, which we term a quality of service
(QoS) offset, can also be added to increase a burst’s priority
and implement priority classes [2]. If the total offset of the
control packet for a class i burst is longer than the offset of a
class j burst plus the maximum class j burst length, then class i
will be isolated from class j [3]. In this paper we consider OBS
networks that employ JET with multiple classes. We further
assume that no buffering or wavelength conversion is available
in the network core.
Analytical models have previously been proposed for OBS
networks with multiple priority classes. In [4] and [3], the
authors assume that the QoS offsets are sufficiently large to
ensure class isolation and that the OBS system obeys the
conservation law.1 This leads to a recursive analytical formula
that approximates the blocking probability of each class using
the Erlang-B blocking formula. However, depending on the
ingress traffic characteristics, the burst aggregation scheme,
and the resulting maximum burst length, achieving class
isolation for even two classes may impose a large delay on
the higher priority class. For a network with N classes, this
imposed delay must be increased N -fold if isolation between
each class is to be achieved. Therefore, depending on the
latency requirements of the traffic, complete isolation may
not be practical. Further, it has been shown that there are a
number of different traffic scenarios for which the conservation
law does not hold [6], so it is desirable to have models that
are accurate for these systems as well. This motivates a more
general model that is accurate for any degree of class isolation
and for systems that do not necessarily obey the conservation
law.
In OBS systems without complete isolation, the degree of
isolation and the resulting performance of the OBS system
1 In general, (work) conservation is a term used to describe scheduling
mechanisms in which the priority discipline does not effect the overall average
performance of the system [5]. If the conservation law holds for an OBS
system, the overall rate of blocking is unaffected by the number of classes
and the size of the QoS offsets assigned to them.
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may depend quite strongly on the burst length distributions.
This, coupled with the fact that burst aggregation mechanisms
and their effect on burst-length distributions is still a very
active area of research [7][8][9][10] motivates the development
of models that are applicable and accurate for arbitrary burst
length distributions.
In this paper, we present an analytical model for computing
blocking probability of a multi-class OBS system. The model
explicitly takes into account the control packet offset and the
burst length distribution. As such, the results are accurate for
an arbitrary number of classes with an arbitrary degree of
class isolation. Further, since these probabilities are computed
directly (i.e. not recursively as in previous models), the results apply equally well to work-conserving and non-workconserving systems. Lastly, by allowing for closed-form, nonrecursive expressions for blocking probabilities, our model
may also be amenable for use in network design and operation.
In Section II and Section III, we derive the model. In
Section IV, we use the model to obtain closed form expressions for blocking probabilities of each class for both the
cases of fixed and exponential burst-length distributions. We
demonstrate the model’s accuracy and examine the performance of OBS networks with and without class isolation in
Section V. Conclusions are drawn and future work is discussed
in Section VI.
II. M ODELING THE T WO -B URST B LOCKING P ROCESS
Since OBS uses advanced reservation, a burst can be
blocked if a contending burst overlaps its head, its tail, both its
head and tail, or some part of its middle, as shown in Fig. 1,
where
• Ti is the arrival time of a given class i control packet
• Li is a random variable representing the length of a class
i burst
• ωi is the offset time between the control packet and
burst for class i (including the control packet length
and assuming that processing-time offset is negligible
compared to QoS offset).
We can represent all four types of blocking shown in Fig. 1
simultaneously, by recognizing that a given class i burst will
be blocked by a class j burst, if all of the following events
occur:
1) the control packet of the class j burst arrives before that
of the class i burst
2) the start of the class j burst arrives before the end of
the class i burst
3) the end of the class j burst arrives after the start of the
class i burst.
We denote the intersection of these three events as Bij . Thus
we have
P [Bij ] = P [(Tj < Ti ) ∩ (Tj + ωj < Ti + ωi + Li )
∩ (Tj + ωj + Lj > Ti + ωi )].
(1)
We define δij ωi − ωj as the offset difference between
class i and class j. If bursts in class j arrive according to
a Poisson process, then we can make use of the memoryless
property [11] to write
P [Bij ] = P [(Tj − Ti ) ∈ [δij − Lj , min(0, Li + δij )]]
= P [τj < min(0, Li + δij ) − (δij − Lj )]
(2)
(3)
where τj is a random variable representing the interarrival time
of class j bursts.
For the multi-class model in the next section, we will require
the probability of Bij , the complement of Bij . From (3), we
have
P [Bij ] = P [τj > min(0, Li + δij ) − (δij − Lj )]

δij < 0, Li < −δij
 P [τj > Lj + Li ],
P [τj > Lj − δij ],
δij < 0, Li −δij
=

P [τj > Lj − δij ],
δij 0
(4)
(5)
By conditioning over all possible values of Li , we can combine
the first two cases in (5) and write
 −δij

 0 P [τj > Lj + li ]fLi (li )dli
∞
P [Bij ] =
+ −δij P [τj > Lj − δij ]fLi (li )dli , δij < 0

 P [τ > L − δ ],
δij 0.
j
j
ij
(6)
 −δij
 0
P [τj > Lj + li ]fLi (li )dli
=
+P [τj > Lj − δij ](1 − FLi (−δij )), δij < 0

P [τj > Lj − δij ],
δij 0
(7)
where fLi (l) and FLi (l) are the probability density and distribution functions for the length of class i bursts respectively,
and P [τj > Lj + li ] and P [τj > Lj − δij ] can be evaluated
using (24) (see appendix).
III. M ULTI -C LASS B LOCKING M ODEL
In this section, without loss of generality, we assume that
the network has N classes of traffic labelled 1, . . . , N such
that δij > 0 for all i < j. Thus, class 1 has the highest
priority, class 2 has the second highest, etc. We are interested
in finding Pbi the average blocking probability of a class i
burst. When a control packet arrives, it attempts to reserve a
time-slot of bandwidth to accommodate its burst. We call this
time-slot the reservation window (RW) of the burst. When a
control packet arrives, if one or more burst lie in its RW, the
bursts is blocked. Let us denote the event that k bursts have
arrived in the reservation window of a class i burst as aki .
Using this definition we have
Pbi = 1 − P [no bursts lie in RW of class i burst]
∞
= 1 − P [a0i ] −
P [aki ∩ all k are blocked]
= 1 − P [a0i ] −
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k=1
∞ k=1
(8)
(9)
P [k bursts blocked | aki ] · P [aki ] .
(10)
Header i
Class i burst
Header j
Tj
Ti
Class j burst
Tj+Ȧj Ti+Ȧi
Tj+Ȧj+Lj Ti+Ȧi+Li
time
We then substitute (7) into (13), to yield
i−1
−δij
P [τj > Lj + li ]fLi (li )dli
Pbi = 1 −
j=1
Header i
Class i burst
Header k
Tk Ti
Ti+Ȧi
Class k burst
Ti+Ȧi+Li
Tk+Ȧk
(b)
time
Ti
Class j burst
Ti+Ȧi+Li
(c)
Tj+Ȧj Ti+Ȧi
time
Tj+Ȧj+Lj
Header i
Tk Ti
Class k burst
Tk+Ȧk+Lk
Ti+Ȧi
Tk+Ȧk
(d)
time
Ti+Ȧi+Li
Fig. 1. Types of Blocking in OBS network. (a) Contending burst overlaps
head of blocked burst. (b) Contending burst overlaps tail of blocked burst. (c)
Contending burst overlaps both head and tail of blocked burst. (d) Contending
burst overlaps middle of blocked burst. Note that (b) and (d) are only possible
if the priority class of the contending burst is higher than that of the blocked
burst (i.e. if ωk > ωi )
For practical networks, blocking events are relatively rare.
(k)
(0)
Thus, P [k bursts blocked | ai ] P [ai ], so we can ignore
all higher order terms in (10), yielding


N
Pbi 1 − P [a0i ] = 1 − P 
Bij 

=1−P 
j=1
i−1
j=1
Bij
P [τj > Lj − δij ].
N

Bij 
Equation (14) gives the expression for the average blocking
probability of a class i burst when there are N classes of
traffic, each with Poisson arrivals, and each with an arbitrary
burst-length distribution and offset time.
IV. E XPONENTIAL AND C ONSTANT B URST L ENGTHS
D ISTRIBUTIONS
Class i burst
Header k
N
(14)
j=i
Class i burst
Tj
·
Tk+Ȧk+Lk
Header i
Header j
0
+ [1 − FLi (−δij )]P [τj > Lj − δij ]
(a)
(11)
We now use the model derived in Section II and III to obtain
closed-form expressions for the blocking probability of each
class in networks with constant and exponential burst-length
distributions.
A. Exponential Burst Lengths
In this section, we assume that the burst lengths for each
class follows an exponential distribution. Thus, the probability
density function fLk and corresponding distribution function
FLk for class k bursts are
fLk (l) =
where we have divided the above intersection into two subsets,
corresponding to the effects of higher-priority traffic, and the
effects of equal-priority and lower-priority traffic. From (7),
we can see that for i j, or equivalently when δij 0, all
of the Bij are independent since each depends only on τj and
Lj . So we have


i−1
N
(12)
Pbi = 1 −
P [Bij ] · P  Bij  .
and
FLk (l) = 1 − e−l/lk
(15)
where lk is the mean burst length of class k. Pbi can be found
directly by substituting (25) and (15) into (14) and solving the
resulting integral to yield
Pbi = 1−
i−1 1+li λj eδij (λj +1/lj ) N j=1
j=i
1 −l/lk
e
lk
1+li λj
N
j=1 [1
j=i
1 + ρj (1 − e−δij/lj )
+ ρj ]
(16)
where λk is the mean arrival rate of class k bursts, and where
ρk = lk λk is the offered load of class k.
B. Fixed Burst Lengths
(13)
Here we assume that all bursts from class i have a fixed
length equal to li and compute Pbi , the blocking probability
for class i. Proceeding directly from (5), and using the fact that
the class i burst interarrival time τi is simply an exponential
random variable with mean 1/λi , we have


e−λj (li +lj ) ,
δij < 0, li < −δij


 −λj (lj −δij )
e
,
δij < 0, li −δij
P [Bij ] =
(17)
−λj (lj −δij )

,
0 δij lj

 e

1,
δij > lj .
accuracy of the resulting model, as described in V, attests to the
validity of this approximation.
If we again define ρi = λi li as the offered load of class i then
we can write
(18)
P [Bij ] = e−ρj (1−dij )
j=1
j=i
When δij < 0, all of the Bij term depend on Li , and thus
independence is not assured. However, in this paper, we make
the approximation2 that they are independent, and write
Pbi = 1 −
i−1
j=1
2 The
P [Bij ]
N
P [Bij ].
j=i
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
 1,
dij =
δ /l ,
 ij j
li /lj ,
1.E-02
δij lj
−li < δij < lj
δij −li
(19)
and dij can be naturally thought of as the degree of isolation
between class i and class j. Finally, substituting (18) into (13),
we have
N
exp
j=1 ρj dij
.
Pbi = 1 −
(20)
N
exp
ρ
j
j=1
Blocking Probability
where
1.E-03
1.E-04
0
V. ACCURACY OF M ODEL AND S IMULATION R ESULTS
A. Modeling the Effects of Different Degrees of Class Isolation
For the results in this section, the overall traffic load (sum
of class 1 and class 2 traffic) was 5 × 10−3 Erlangs, and the
ratio of class 1 (higher class) traffic to class 2 traffic was 0.1.
An expression for the blocking probability for two classes
with exponential burst lengths was found by setting N = 2 in
(16), yielding
Pb1 = 1 −
Pb2 = 1 −
1 + ρ2 (1 − e−δ12 /l2 )
(1 + ρ1 )(1 + ρ2 )
δ21 (λ1 +1/l1 )
1 + l2 λ 1 e
.
(1 + ρ1 )(1 + ρ2 )(1 + l2 λ1 )
(21)
Similarly, for the case of fixed-length bursts, we set N = 2
in (20) to yield
eρ2 d12
eρ1 d21
,
P
=
1
−
.
(22)
b2
eρ1 +ρ2
eρ1 +ρ2
Fig. 2 compares the simulation results to our model and
a model that assumes conservation between classes [4]. CL1
and CL2 refer to class 1 and class 2 respectively. We plot the
blocking probability for each class as we vary the QoS offset
of class 1 traffic, while keeping the class 2 QoS offset equal
to zero. The results for fixed and exponential length burst are
shown in Fig. 2(a) and Fig. 2(b) respectively. On the abscissa,
we show the normalized QoS offset difference, ∆12 = δ12 /l2 ,
which we have defined as the ratio of the difference of the
class 1 and class 2 offsets to the mean burst length of class 2.
Pb1 = 1 −
Complete isolation is achieved for fixed-length bursts when
∆12 1. Beyond this point, the blocking probabilities are
constant. For the exponential case, there is no maximum burst
length, so complete isolation cannot be achieved. However, for
the parameters simulated, the two classes are nearly isolated
0.2
0.4
0.6
0.8
1
1.2
Normalized Qos offset difference ( ǻ12 = į 12/l 2 )
1.4
1.6
(a)
1.E-02
Blocking Probability
We simulated a two-class OBS node to demonstrate the
accuracy of the analytical model and examine the effects
of offered load and offsets on network performance. For
each simulation, depending on the offered load, between two
million and one hundred million bursts were simulated. Bursts
arrived according to a Poisson process with an average burst
length of 10 time units. Two different burst-length distributions
were simulated: exponential and fixed-length. We found that
our model was accurate for any degree of class isolation and
over a large range of offered loads.
CL2, simulation
CL2, Barakat & Sargent
CL2, Yoo & Qiao
CL1, simulation
CL1, Barakat & Sargent
CL1, Yoo & Qiao
CL2, simulation
CL2, Barakat & Sargent
CL2, Yoo & Qiao
CL1, simulation
CL1, Barakat & Sargent
CL1, Yoo & Qiao
1.E-03
1.E-04
0
1
2
3
4
5
6
Normalized Qos offset difference ( ǻ12 = į 12 /l 2 )
7
8
(b)
Fig. 2. Blocking probability versus QoS offset of high class for two-class
OBS node. Bursts’ lengths are fixed in (a) and exponentially distributed in
(b).
for ∆12 > 8. Subsequent simulations showed that value is
dependent on not only the burst-length distribution, but also on
the relative loads of the higher and lower class traffic. Thus, the
nature of the lower class traffic must be carefully considered
when selecting the size of the QoS offsets to achieve isolation.
For high values of ∆12 , when class 1 is well isolated from
class 2, both models are quite accurate. However, as ∆12
decreases and class 1 becomes less isolated from class 2,
the blocking experienced by class 1 and class 2 increases
and decreases respectively. Because the model in [4] assumes
complete isolation between classes, its computed blocking
probabilities remain constant and are inaccurate for small
values of ∆12 , deviating from the simulation results by as
much as an order of magnitude. By contrast, the model
presented in this paper makes no such assumption, and its
output curve agrees closely with the simulation for all values
of ∆12 . This ability to model accurately the blocking behavior
in the regime of low ∆12 is particularly useful for systems
with long bursts or low latency requirements, where complete
isolation may not be achievable.
B. Effect of Offered Load on Accuracy of Model
In this section, we examine the accuracy of our analytical
model over a large range of offered loads. We considered a
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−δ13 /l3
)][1 + ρ3 (1 − e
[1 + ρ2 (1 − e
(1 + ρ1 )(1 + ρ2 )(1 + ρ3 )
1.E+00
)]
Pb2
[1 + ρ3 (1 − e−δ23 /l3 )][1 + l2 λ1 eδ21 (λ1 +1/l2 ) ]
=1−
(1 + ρ1 )(1 + ρ2 )(1 + ρ3 )(1 + l2 λ1 )
Pb3
[1 + l3 λ1 eδ31 (λ1 +1/l3 ) ][1 + l3 λ2 eδ32 (λ2 +1/l3 ) ]
=1−
.
(1 + ρ1 )(1 + ρ2 )(1 + ρ3 )(1 + l3 λ1 )(1 + l3 λ2 )
(23)
Fig. 3(a) shows results when the offset of class 1, class 2,
and class 3 were set to 0, 30, and 60 respectively. For the
results in Fig. 3(b), the offsets for class 1, class 2, and class
3 were 0, 20, and 40 respectively. Thus, Fig. 3(a) corresponds
to a network in which each class is well isolated from the
classes below it, and Fig. 3(b) corresponds to a network for
which classes are not isolated from one another.
In both sets of simulation results in Fig. 3, we see that the
relative priority and relative blocking probability of each class
remains unchanged as the load varies up to 1, above which,
the blocking probability of all classes begins to asymptotically
approach 1. Therefore, we conclude that the QoS offset mechanism for class differentiation scales very well to arbitrary
traffic loads.
In both Fig. 3(a) and Fig. 3(b), for offered loads < 1, our
model predicts the blocking probability almost exactly for all
three classes. For offered loads 1, the predicted blocking
probability deviates slightly from the simulation. This is due
to the first order approximation in (11), which assumes that the
blocking rate in then network 1. When the overall offered
load is large enough such that the resulting blocking rate is
greater than 0.1, the above assumption does not hold, and the
predictions from our model begin to deviate slightly from the
simulations. However, this deviation is quite small even as the
blocking rates approach 1. Thus, we conclude that, although
our model is very accurate for all levels of blocking, it is
most well suited for networks in which the blocking rate is
less than 10%, which includes the vast majority of practical
communication networks.
VI. C ONCLUSIONS
In this paper, we derived an analytical model that evaluates
the blocking probability for each class in a multi-class OBS
system with arbitrary burst length distributions. Using simulation, we showed that the model accurately computes the
average blocking probability experienced by multiple classes
of traffic for both the cases of isolated and non-isolated traffic
classes. The model was also shown to be very accurate for all
values of offered load simulated.
1.E-01
1.E-02
CL3, simulation
CL3, Barakat & Sargent
CL2, simulation
CL2, Barakat & Sargent
CL1, simulation
CL1, Barakat & Sargent
1.E-03
1.E-04
1.E-05
1.E-06
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Overall Network Load
(a)
1.E+01
1.E+00
Blocking Probability
Pb1 = 1 −
−δ12 /l2
1.E+01
Blocking Probability
network with three classes of traffic and exponential burstlength distributions. We fixed the ratio of Class 2 traffic to
Class 1 traffic and the ratio of class 3 traffic to class 2 traffic
at 10, while varying the overall offered load from 10−4 to 103 .
A closed-form expression for the blocking probability of
each class when burst lengths are exponentially distributed
was found by setting N = 3 in (16) to yield
1.E-01
1.E-02
CL3, simulation
CL3, Barakat & Sargent
CL2, simulation
CL2, Barakat & Sargent
CL1, simulation
CL1, Barakat & Sargent
1.E-03
1.E-04
1.E-05
1.E-06
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Overall Network Load
(b)
Fig. 3. Blocking probability versus overall network load with three classes.
Offsets are chosen such that each class is isolated from lower classes in (a)
and not isolated from lower classes in (b).
Future work includes extending the model to include the
effects of wavelength conversion. Further, the model’s ability
to derive closed-form expressions for blocking implies that
it can be used in future studies to aid in network design
and operation. For example, by inverting the expressions in
Section IV, offsets could be chosen based on desired blocking
rates of each class, and traffic policing could be performed at
the edge of the network to ensure that these blocking rates are
achieved.
A PPENDIX
If X and Y are non-negative random variables with joint
distribution fXY (xy) and c is a constant,
∞ x−c
0∞ 0∞ fXY (x, y)dy dx c < 0
P [X > Y + c] =
f (x, y)dx dy c 0.
0
y+c XY
(24)
If X and Y are independent exponential random variables
with mean 1/ax and 1/ay respectively,
P [X > Y + c] =
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ay +ax (1−eay c )
ay +ax
ay e−ax c
ay +ax
c<0
c 0.
(25)
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